Abstract
In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.
Highlights
The dynamics of a two-level system interacting with a weak electromagnetic field is a relevant field of study in quantum optics
Slow oscillating terms are often removed by Rotating wave approximation (RWA) in some models
Is it fast enough to be neglected? Despite RWA is a “rude” method, it usually gives a good approximation for some important models in quantum optics
Summary
The dynamics of a two-level system interacting with a weak electromagnetic field is a relevant field of study in quantum optics. To our knowledge, it is not clear how to get the high-order approximate solutions rigorously for the quasi-resonant control. We develop a high-order RG method for Schrödinger equations with quasi-resonant control. The dynamic behavior of two-level systems near resonance is the most concerned problem, which is the original intention of developing the high-order RG method. This paper introduces a time transformation technique to avoid the invalid expansion near resonance. The numerical simulations are presented to illustrate that the first-order RG approximate solution is usually good enough in the high-order near resonant case, but the second-order RG approximate solution plays a fundamental role in the two-scale case and high-order weak driving case. High-order RG approximate solutions in several types of near resonance are discussed in detail
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